What are the constituents of communicated time differences between moving objects?

In "The Elegant Universe," two individuals, George and Gracie, are noted to be moving relative to one another in space and wearing clocks. They sync their clocks upon passing each other. Each observes the other's clock to be ticking more slowly. Gracie communicates with George via cell.

Gracie communicates her time to George as he recedes into the distance. Initially, Gracie thinks that George will hear her reported time before his clock reaches the same time. But then, she takes into account travel time and realizes the travel time will more than compensate, and George will actually hear her time after his clock has passed that time.

Then, the author notes that

Gracie realizes that even if George takes the travel time into account, he will conclude from Gracie's communication that her clock is running slower than his.

By this last statement, does it mean that the time difference will be greater than what can be accounted for by mere communication travel time? I would think that some of the time difference is due to Gracie (from George's perspective) transmitting the time after that time has already elapsed on his clock. After all, Gracie's clock is moving more slowly (due to time dilation) relative to his clock and that accounts for the extra time difference (beyond the communication travel time). Do I have that correct?

In George's frame Gracie passes him at some velocity $v$, so George can calculate the distance to Gracie as a function of time. That means when George receives a time signal from Gracie he can calculate how long the signal took to reach him (travelling at $c$). If George adds this travel time to the time in Gracie's time signal the result will be a time that does not match the time on George's clock. From this George will know that Gracie's clock must be running at a different rate to his.